Prove that:
(i) tan20°tan35°tan45°tan55°tan70°=1
(ii) sin48°sec42°+cos48°cosec20°=2
(iii) sin70°/cos20°+cosec20°/sec70°-2cos70°cosec20°=0
(iv) cos80°/sin10°+cos59°cosec31°=2
Answers
Answer:
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All proven
Step-by-step explanation:
(i) Prove tan20° tan35° tan45° tan55° tan70° = 1
LHS = tan20° tan35° tan45° tan55° tan70°
= tan20°. tan 70°. tan35° .tan55°. tan45°
= tan20°. Cot(90-70)° .tan35°. Cot(90-55). tan45°
= tan20°. Cot20°. tan35°. cot35°. tan45°
= 1 * 1 * 1
= 1
LHS = RHS. Hence proved.
(ii) Prove sin48° sec42° + cos48° cosec32° = 2
LHS = sin48° sec42°+ cos48° cosec20°
= Sin48°. Cosec(90-42)° + cos48°. Sec(90-32)°
= Sin48°. Cosec48° + Cos48°. Sec48°
= 1 + 1
= 2
LHS = RHS. Hence proved.
(iii) Prove sin70° / cos20° + cosec20° / sec70° -2cos70° cosec20° = 0
LHS = sin70°/ cos20° + cosec20°/ sec70°- 2cos70° cosec20°
= sin70°/ Sin70° + cosec20°/ cosec20° - 2Cos70. Sec70°
= 1 + 1 - 2(1)
= 0
LHS = RHS. Hence proved.
(iv) Prove cos80°/ sin10° + cos59° cosec31° = 2
LHS = cos80°/ sin10°+ cos59° cosec31°
= Cos80°/ cos80° + Cos59°. Sec59°
= 1 + 1
= 2
LHS = RHS. Hence proved.